Prime number

Here is another way to think of prime numbers. The number 12 is not prime, because a rectangle can be made, with sides of lengths 4 and 3. This rectangle has an area of 12, because all 12 blocks are used. This cannot be done with 11. No matter how the rectangle is arranged, there will always be blocks left over, except for the rectangle with sides of lengths 11 and 1. 11 must therefore be a prime number.

A prime number is a natural number of a particular kind. Any natural number is equal to 1 times itself. If the number is equal to any other natural numbers multiplied, then the number is called a composite number. The smallest composite number is 4, because 2 x 2 = 4. 1 is not a composite number. Every other number is a prime number. The prime numbers are the numbers other than 1 which are not equal to [math]\displaystyle{ m \times n }[/math] (except 1 times itself). The smallest prime number is 2. The next prime numbers are 3, 5, 7, 11, and 13. There is no largest prime number. The set of prime numbers is sometimes written as [math]\displaystyle{ \mathbb{P} }[/math].

The fundamental theorem of arithmetic states that every positive integer can be written as a product of primes in a unique way, though the way the prime numbers occur is a difficult problem for mathematicians. When a number is larger, it is more difficult to know if it is a prime number. One of the answers is the prime number theorem. One of the unsolved problems is Goldbach's conjecture.

One of the most famous mathematicians of the classical era, Euclid, recorded a proof that there is no largest prime number. However many scientists and mathematicians are still searching to find it as part of the Great Internet Mersenne Prime Search.

How to find small prime numbers

There is a simple method to find a list of prime numbers. Eratosthenes created it. It has the name Sieve of Eratosthenes. It catches numbers that are not prime (like a sieve), and lets the prime numbers pass through.

The method works with a list of numbers, and a special number called b that changes during the method. As one goes through the method, they circle some numbers in the list and cross out others. Each circled number is prime and each crossed-out number is composite. At the start, all the numbers are plain: not circled and not crossed out.

The method is always the same:

1. On a sheet of paper, write all the whole numbers from 2 up to the number being tested. Do not write down the number 1. Go to the next step.
2. Start with b equal to 2. Go to the next step.
3. Circle b in the list. Go to the next step.
4. Starting from b, count up b more in the list and cross out that number. Repeat counting up b more numbers and crossing out numbers until the end of the list. Go to the next step.
   • (For example: When b is 2, you will circle 2 and cross out 4, 6, 8, and so on. When b is 3, you will circle 3 and cross out 6, 9, 12, and so on. 6 and 12 have already been crossed out. Cross them out again.)
5. Increase b by 1. Go to the next step.
6. If b has been crossed out, go back to the previous step. If b is a number in the list that has not been crossed out, go to the 3rd step. If b is not in the list, go to the final step.
7. (This is the final step.) You are done. All of the prime numbers are circled and all of the composite numbers are crossed out

For example, one could carry out this method on a list of the numbers from 2 to 10. At the end, the numbers 2, 3, 5, and 7 will end up circled. These are prime numbers. The numbers 4, 6, 8, 9 and 10 will be crossed out. These are composite numbers.

This method or algorithm takes too long to find very large prime numbers. However, it is less complicated than methods used for very large primes, such as Fermat's primality test (a test to see whether a number is prime or not) and the Miller-Rabin primality test.

What prime numbers are used for

Prime numbers are very important in mathematics and computer science. Very long numbers are hard to solve. It is difficult to find their prime factors, so most of the time, numbers that are probably prime are used for encryption and secret codes. For example:

• Most people have a bank card, where they can get money from their account using an ATM. This card is protected by a secret access code. Since the code needs to be kept secret, it cannot be stored in cleartext on the card. Encryption is used to store the code in a secret way. This encryption uses multiplications, divisions, and finding remainders of large prime numbers. An algorithm called RSA is often used in practice. It uses the Chinese remainder theorem.

• If someone has a digital signature for their email, encryption is used. This makes sure that no one can fake an email from them. Before signing, a hash value of the message is created. This is then combined with a digital signature to produce a signed message. Methods used are more or less the same as in the first case above.

• Finding the largest known prime number has, over the years, become a sport of sorts. Testing if a number is prime can be difficult if the number is large. The largest primes known at any time are usually Mersenne primes, because the fastest known test for primality is the Lucas-Lehmer test, which relies on the special form of Mersenne numbers.

Prime Number Media

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  Composite numbers can be arranged into rectangles but prime numbers cannot.

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  Demonstration, with Cuisenaire rods, that 7 is prime, because none of 2, 3, 4, 5, or 6 divide it evenly

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  The Rhind Mathematical Papyrus

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  The relative error of \tfrac{n}{\log n} and the logarithmic integral \operatorname{Li}(n) as approximations to the prime-counting function. Both relative errors decrease to zero as n grows, but the convergence to zero is much more rapid for the logarithmic integral.

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  The Ulam spiral. Prime numbers (red) cluster on some diagonals and not others. Prime values of 4n^2 - 2n + 41 are shown in blue.

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  Plot of the absolute values of the zeta function, showing some of its features

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  The Gaussian primes with norm less than 500

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  The small gear in this piece of farm equipment has 13 teeth, a prime number, and the middle gear has 21, relatively prime to 13.

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  The sieve of Eratosthenes starts with all numbers unmarked (gray). It repeatedly finds the first unmarked number, marks it as prime (dark colors) and marks its square and all later multiples as composite (lighter colors). After marking the multiples of 2 (red), 3 (green), 5 (blue), and 7 (yellow), all primes up to the square root of the table size have been processed, and all remaining unmarked numbers (11, 13, etc.) are marked as primes (magenta).

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  The connected sum of two prime knots

Related pages

• Coprime
• List of prime numbers
• Palindromic prime
• Prime factorization
• Wilson prime

Other websites

• GIMPS, a group that searches for Mersenne primes